A particle located at `x = 0` at time `t = 0`, starts moving along with the positive x-direction with a velocity v that varies as ` v = alpha sqrt(x)`. The displacement of the particle varies with time as

Options

A. t
B. `t^(1//2)`
C. `t^(3)`
D. `t^(2)`

Answer

D

Explanation

`upsilon=alpha sqrt(x) rArr (dx)/(dt) = alpha sqrt(x)`
`rArr (dx)/(sqrt(x))=alpha dt`
By integrating both sides
`int x^(-1//2)dx= int alpha dt`
`(sqrt(x))/(1//2)=alpha t rArr sqrt(x)=(1)/(2)alpha t`
`rArr x=(1)/(4)alpha^(2)t^(2)`
`therefore x prop t^(2)`